Singular Value Consulting

Flying to Mars in three days

Richard Campbell brought up an interesting idea in his recent Mars geek out show. Suppose you could travel to Mars accelerating at 1 g for the first half the trip, then decelerating at 1 g for the final half of the trip. Along the way you’d feel a force equal to the force of gravity you’re used to, and you’d get there quickly. How quickly? According to the show, just three days.

To verify this figure, we’ll do a very rough calculation. Accelerating at 1 g for time t covers a distance is g t2/2. Let d be the distance to Mars in meters, T the total of the trip in seconds, and g = 9.8 m/s2. In half the trip you cover half the distance, so 9.8 (T/2)2/2 = d/2. So T = 0.64 √d.

The hard part is picking a value for d. To keep things simple, assume you head straight to Mars, or rather straight toward where Mars will be by the time you get there. (In practice, you’d take more of a curved path.) Next, what do you want to use as your straight-line distance? The distance between Earth and Mars varies between about 55 million km and 400 million km. That gives you a time T between 1.7 and 4.7 days.

We don’t have the technology to accelerate for a day at 1 g. As Richard Campbell points out, spacecraft typically accelerate for maybe 20 minutes and coast for most of their journey. They may also pick up speed by slinging around a planet, but there are no planets between here and Mars.

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20 thoughts on “Flying to Mars in three days”

This idea has been kicking around for a long time. I have a feeling Isaac Asimov, or Arthur C Clarke or perhaps Larry Niven mentioned it once decades ago. Anyway here’s an article from a few years ago which confirms the maths http://uk.answers.yahoo.com/question/index?qid=20100430122358AAcPffc (scroll to last answer). On-board nuclear fusion should be more than adequate!

Nice calculation…
Now let’s see why it’s not possible with current technology… especially how much energy you’d need for such a trip.
Let’s imagine you+spacecraft weigh 100kg (very advanced tech, I know) and you take 1.7 days to go there.
Needed force = 100N
Max speed = 73440 m/s
Avg speed = 36720 m/s (since acceleration is constant)
Avg power needed = 3,672 kW
Total needed energy = 539,4 GJ
Assuming you don’t have any loss of energy, it would take the amount of energy contained in 4,5 tons of hydrogen.. or 7 grams of pure U235.

A conventionnal chemical proppeller would surely not be enough for that!

There is a very interesting variant to this problem posed to me as an exercise of relativity. It is inspired in the following fact, to sustain 1g acceleration for the whole journey one will probably need a massive storage of energy, and that is precisely mass to energy conversion. So the question is, what is the total (initial) mass of the rocket in order to carry a one capsule of one ton to the closest start at 1g, assuming that mass can be converted into energy with a 100% efficiency. (The 1g constrain is just so the humans seat comfortably). Note that the last half of the trip is a -1g deceleration.

But would the fuel always have to be lifted from the surface of the earth? If I were writing this story I think I’d have a robotic orbital infrastructure harvesting fuel from Titan (moon of Jupiter) and shipping it in gigantic freighters to somewhere fairly nearly earth. Yes that all takes a lot of time and money and so on and so forth, but at some point after all of that, a person leaves the surface of the earth in some conventional vehicle (SpaceX or whatever) and only boards a Mars-capable vehicle in orbit. Then he can have all the fuel he needs. Solving the “harvesting fuel from Titan” problem is left as an exercise for the reader – at least the robots don’t care how long it takes.

Yes you could do something like this and it is contemplated as “cryogenic depots”, the idea, at the very least would be to lift this from earth whenever the cargo of a SpaceX vehicle allows for that (sor sort of dummy payload). The clear issue is not the idea of a space depot or how to fill them, the issue is that currently, our technology does not allow for these depots to keep these cryogens for long periods of time. In fact, right now it is difficult to keep a hydrogen tank for more than a day without releasing it to space due to boil-off.

Ed: you beat me to the punch; I was also about to mention the other planets that are (sometimes) between Earth and Mars.

Chris Morgan: Good idea. Rather than shuttling all that fuel to one location somewhere near Earth, it would be slightly more efficient to divide that fuel up among a series of “refilling stations” between Earth and Mars, and only send just enough fuel to Earth for the ship to get to the first refilling station.

Igor Carron: Hydrogen is the best fuel for launching stuff from dirtside to orbit. But because of various reasons — and your point is perhaps the most important one — it may be better to stockpile some other reaction mass at those refilling stations — rocks and dust and ice.

I did a little calculation, and it seems that Peter above is incorrect. Average velocity on this trip would be 435 kilometers per second, with a max speed of around 880 kilometers per second at the point where it turns over and begins decelerating. Hitting a 1-gram grain of sand at that speed would create an impact of 700-1500 kJ, which would be enough to destroy your ship.

A materials engineer in the aerospace engineering department at NCSU once told me that this constant 1G acceleration idea is great from a propulsion standpoint, and from the standpoint of stress on the superstructure (which no longer has to endure 5-8G stresses for take-off), but that unless you can design a stronger material from which to create your hull, a grain of sand will destroy your entire enterprise.

@Chredon, you work under the assumption that the efficiency of the inelastic collision will be 100%, in my opinion this is far from the real case. There exists the likely possibility of off-head on collision in which case the grain material is deflected or even in a head on collision that the grain of sand would transverse the whole ship without releasing all or even most of the energy, leaving a track that can mostly self heal with resolidification.

Long distance ships will need final assembly in space. Conventional rockets can put large pieces in orbit, the earth will be the first slingshot, perhaps burning a conventional rocket for the early acceleration. Whatever is most effective, the trip will need to be made by a plutonium fuel system though. We have ships and subs all over the world on reactor power, only challenge really is cooling in space. The particle collision is a issue, but a depleted uranium forward dust shield will minimize this and perhaps even a occasional pulse from a charged particle beam to get the ones coming on dead canter. This ship cannot be the tinfoil lunar lander used for moon shots.